Problem 10 - University Physics with Modern Physics

Question

If you look at an object at the bottom of a pool, the pool looks less deep than it actually is. a) From what you have learned, calculate how deep a pool seems to be if it is actually 4 feet deep and you look directly down on it. The refractive index of water is \(1.33 .\) b) Would the pool look more or less deep if you look at it from an angle other than vertical? Answer this qualitatively, without using an equation.

Step-by-Step Solution

Answer

a) The apparent depth of the pool when looking straight down into it will be approximately 3 feet. b) The pool would appear deeper when viewed from an angle other than vertical.

Step 1: Snell's Law

Snell's Law is given by the equation \(n_1 \sin{\theta_1} = n_2 \sin{\theta_2}\), where \(n_1\) and \(n_2\) are the refractive indices of the two media (in this case, air and water) and \({\theta_1}\) and \({\theta_2}\) are the angles of incidence and refraction, respectively. From the problem, we know that \(n_1 = 1\) and \(n_2=1.33\), and we are looking straight down into the water, so \(\theta_1 = 0\). We can then find \(\theta_2\) from Snell's Law. Step 2: Find the apparent depth of the pool

Step 2: Calculate Apparent Depth

To find the apparent depth, we can use the equation \(d_{apparent} = d_{actual} \frac{\sin{\theta_1}}{\sin{\theta_2}}\), where \(d_{apparent}\) is the apparent depth, \(d_{actual}\) is the actual depth, and \({\theta_1}\) and \({\theta_2}\) are the angles we found in step 1. We plug in the values for \(d_{actual} = 4\) feet and the angles to find the apparent depth. Step 3: Determine whether the apparent depth changes when viewed from an angle

Step 3: Viewed from an Angle

The last part of the problem asks whether the apparent depth would be more or less when viewed from a non-vertical angle. We can answer this qualitatively without equations by considering how light refracts as it enters the water. When viewed vertically, the light refracts directly upward, making the pool appear shallower. As the angle of incidence increases, the light refracts more, causing the angle between the observer and the bottom of the pool to change, which makes the pool seem deeper. a) When solving the steps above, it will give us the apparent depth (shallower) of the pool when looking straight down into it. b) The pool would look deeper (more depth) when viewed from an angle other than vertical.

Key Concepts

Snell's Law

Understanding how light bends, or refracts, when it moves from one medium to another, such as from air into water, is key to understanding why a pool appears shallower than it really is. This bending is accurately described by Snell's Law, which states that the product of the refractive index of the first medium and the sine of the angle of incidence is equal to the product of the refractive index of the second medium and the sine of the angle of refraction. Mathematically, this is represented as
\( n_1 \sin(\theta_1) = n_2 \sin(\theta_2) \).

In simpler terms, Snell's Law helps us predict the path light will take as it enters a new medium, which radically changes what we see. The greater the difference between the refractive indices, the more light is bent. In the context of a swimming pool, this law allows us to calculate the angle at which light rays rise to our eyes from the underwater object and how this angle changes our perception of depth.

Refractive Index

At the heart of Snell's Law is the refractive index, symbolized by 'n', which measures how much a substance can bend light. The refractive index of a medium is a dimensionless number that describes how fast light travels through the medium compared to the speed of light in a vacuum. For instance, water has a refractive index of about 1.33, meaning light travels 1/1.33 times slower in water than in a vacuum.

A higher refractive index indicates that light slows down more and bends more sharply when entering the medium from a vacuum or air. This concept is crucial in explaining why objects under water look closer to the surface than they actually are – the light rays from these objects bend as they exit the water, altering our perception of where the objects are located.

Angle of Incidence

The angle of incidence is defined as the angle between the incoming light ray and an imaginary line, called the normal, which is perpendicular to the surface at the point of contact. It's one half of the puzzle solved by Snell's law. When looking straight down into a pool, the angle of incidence is 0 degrees, since the light travels directly from the object to your eyes.

This angle is crucial in calculating the apparent depth of the pool because, when light enters at different angles of incidence (other than vertically), the amount of bending that the light undergoes changes, affecting how we perceive the depth.

Angle of Refraction

Complementing the angle of incidence is the angle of refraction, which is the angle between the refracted light ray and the normal. According to Snell's Law, this angle can be calculated once we know the refractive indices and the angle of incidence. The angle of refraction determines the direction in which light continues to travel in the new medium.

Understanding the angle of refraction is critical when considering the way we perceive depth in water. When looking straight down into water (a 0-degree angle of incidence), the angle of refraction is also relatively small, leading to a minor distortion in perceived depth. However, as the viewing angle becomes oblique, the angle of refraction increases, and so does the apparent depth misconception, making us think that the pool is deeper.